We consider the evolution of the bulk bubble-size distribution $N(a,t)$ of large bubbles (Weber number ${\textit {We}}\gg 1$) under free-surface entrainment described generally by an entrainment size distribution $I(a)$ with power-law slope $\gamma$ and large-radius cutoff $a_{max}$. Our main focus is the interaction between turbulence-driven fragmentation and free-surface entrainment, and, for simplicity, we ignore other mechanisms such as degassing, coalescence and dissolution. Of special interest are the equilibrium bulk distribution $N_{eq}(a)$, with local power-law slope $\tilde {\beta }_{eq}(a)$, and the time scale $\tau _c$ to reach this equilibrium after initiation of entrainment. For bubble radii $a\ll a_{max}$, we find two regimes for the dependence of $N_{eq}(a)$ on the entrainment distribution. There is a weak injection regime for $\gamma \ge -4$, where $\tilde {\beta }_{eq}(a)=-10/3$ independent of the entrainment distribution; and a strong injection regime for $\gamma <-4$, where the power-law slope depends on $\gamma$ and is given by $\tilde {\beta }_{eq}(a)=\gamma +2/3$. The weak regime provides a general explanation for the commonly observed $-10/3$ power law originally proposed by Garrett et al. (J. Phys. Oceanogr., vol. 30 (9), 2000, pp. 2163–2171), and suggests that different weak entrainment mechanisms can all lead to this result. For $a\sim a_{max}$, we find that $N_{eq}(a)$ exhibits a steepening deviation from a power law due to fragmentation and entrainment, similar to what has been observed, but here absent other mechanisms such as degassing. The evolution of $N(a,t)$ to $N_{eq}(a)$ is characterised by the critical time $\tau _c \propto C_f \varepsilon ^{-1/3} {a_{max}}^{2/3}$, where $\varepsilon$ is the turbulence dissipation rate and $C_f$ is a new constant that quantifies the dependence on the daughter size distribution in a fragmentation event. For typical breaking waves, $\tau _c$ can be quite small, limiting the time $t\lesssim \tau _c$ when direct measurement of $N(a,t)$ might provide information about the underlying entrainment size distribution.

In this experimental study, multiscale rough surfaces with regular (cuboid) elements are used to examine the effects of roughness-scale hierarchy on turbulent boundary layers. Three iterations have been used with a first iteration of large-scale cuboids onto which subsequent smaller cuboids are uniformly added, with their size decreasing with a power-law as the number increases. The drag is directly measured through a floating-element drag balance, while particle image velocimetry allowed the assessment of the flow field. The drag measurements revealed the smallest roughness iteration can contribute to nearly 7 $\%$ of the overall drag of a full surface, while the intermediate iterations are responsible for over $12\,\%$ (at the highest Reynolds number tested). It is shown that the aerodynamic roughness length scale between subsequent iterations varies linearly, and can be described with a geometrical parameter proportional to the frontal solidity. Mean and turbulent statistics are evaluated using the drag information, and highlighted substantial changes within the canopy region as well as in the outer flow, with modifications to the inertial sublayer (ISL) and the wake region. These changes are shown to be caused by the presence of large-scale secondary motions in the cross-plane, which itself is believed to be a consequence of the largest multiscale roughness phase (spacing between largest cuboids), shown to be of the same order of magnitude as the boundary-layer thickness. Implications on the classical similarity laws are additionally discussed.

Compound particles are a class of composite systems in which solid particles encapsulated in a fluid droplet are suspended in another fluid. They are encountered in various natural and biological processes, for e.g. nucleated cells, hydrogels, microcapsules etc. Generation and transportation of such multiphase structures in microfluidic devices is associated with several challenges because of the poor understanding of their structural stability in a background flow and the rheological characteristics of their dispersions. Hence, in this work, we analyse the flow in and around a concentric compound particle and investigate the deformation dynamics of the confining drop and its stability against breakup in imposed linear flows. In the inertia-less limit (Reynolds number, $Re \ll 1$) and assuming that the surface tension force dominates the viscous forces (low capillary number, $Ca$, limit), we obtain analytical expressions for the velocity and pressure fields up to ${O}(Ca)$ for a compound particle subjected to a linear flow using a domain perturbation technique. Simultaneously, we determine the deformed shape of the confining drop correct up to ${O}(Ca^2)$, facilitating the following. (i) Since ${O}(Ca^2)$ calculations account for the rotation of the anisotropically deformed interface, the reorientation dynamics of the deformed compound particles is determined. (ii) Calculations involving the ${O}(Ca^2)$ shape of the confining interface are found to be important for compound particles as ${O}(Ca)$ calculations make qualitatively different predictions in generalised extensional flows. (iii) An ${O}(Ca)$ constitutive equation for the volume-averaged stress for a dilute dispersion of compound particles was developed to study both shear and extensional rheology in a unified framework. Our analysis shows that the presence of an encapsulated particle always enhances all the measured rheological quantities such as the effective shear viscosity, extensional viscosity and normal stress differences. (iv) Moreover, linear viscoelastic behaviour of a dilute dispersion of compound particles is characterised in terms of complex modulus by subjecting the dilute dispersion to a small-amplitude oscillatory shear (SAOS) flow. (v) Various expressions pertaining to a suspension of particles, drops, and particles coated with a fluid film are also derived as limiting cases of compound particles.

Misaligned wind turbine rotors redirect the wake, and manipulate the wake shape by introducing a counter-rotating vortex pair. This mechanism is of great interest for improving wind farm power output through static or dynamic misalignment. In this study, cross-plane stereo-particle image velocimetry measurements are used to characterize the wake evolution for tilt misalignment and verify differences with yaw misalignment. Blockage from the ground, shear in the velocity profile, turbulence levels, hub-vortices and tip-vortices are found to strongly affect wake evolution for a tilted wind turbine resulting in a non-symmetric behaviour for upwards deflecting or downwards deflecting tilt. The downwards deflection of a negatively tilted wind turbine is found to result in the most benefits for wake recovery and power availability downstream through increased wake-curling, faster wake-recovery, and downdraft of high-momentum flow.

This paper studies the hydraulic control of continental shelf waves using an inviscid barotropic quasi-geostrophic model with piecewise-constant potential vorticity, in which the shelf is represented by a flat step of variable width. A coastal-intensified geostrophic current generates topographic Rossby waves, which can become critical at a local decrease in shelf width when the background current opposes Rossby wave propagation. That is, the shelfbreak perturbation permanently modifies the flow field over arbitrarily large distances and the flow transitions from subcritical to supercritical as it crosses the perturbation. Critically controlled flows also lead to the exchange of significant volumes of water between the shelf and the deep ocean. We derive the boundaries for which critical control occurs in terms of a Froude number and the dimensionless magnitude of the perturbation, and analyse the possible transitions between controlled and far-field flow. When first-order dispersive terms are included in the model, transitions are resolved by dispersive shock waves, which remain attached to the forcing region when the Froude number is close to the boundary for critical flow. Contour dynamic simulations show that the dispersive long-wave model captures the quantitative behaviour of the full quasi-geostrophic system for slowly varying shelves, and replicates the qualitative behaviour even when the long-wave parameter is order one.

This paper presents low-order models of Rayleigh–Bénard convection of a radiating gas in a cubic cell, in the Rayleigh number range $Ra \in [ 10^6\text {--}10^8 ]$. Numerical simulations are carried out for an air–$\textrm {H}_2\textrm {O}$–$\textrm {CO}_2$ mixture assumed to be radiating (coupled case) or transparent (uncoupled case). When coupling with radiation, it is shown that the kinetic energy of the flow and the thermal energy increase. At $Ra=10^6$, planar flow states are observed when radiation is taken into account, while diagonal flow states prevail in the uncoupled case. From $Ra\ge 3\times 10^7$, quasi-stable diagonal flows are observed in both coupled and uncoupled simulations, with occasional brief reorientations. The reorientation frequency seems to decrease with the Rayleigh number and seems to increase with radiation. A proper orthogonal decomposition (POD) analysis reveals that 11 of the first 12 POD eigenfunctions are preserved over the Rayleigh number range, whatever the coupling conditions. However, POD eigenvalues are higher with radiation. POD-based low-order models are derived at different Rayleigh numbers, for both coupled and uncoupled cases. Radiative transfer effects are added in the model in an a priori fashion, from uncoupled simulation data. Coupled POD models predict the energy increase with radiation and the loss of stability of the diagonal rolls at $Ra=10^6$. Uncoupled and coupled models correctly reproduce reorientation frequencies over the Rayleigh number range. Finally, a generalised model is derived, solely based on uncoupled simulation data at $Ra=10^7$ and energy scaling laws. This generalised model captures the change in dynamics associated with radiation effects and variations in Rayleigh number, except at $Ra=10^6$.

Thermally driven vertical convection (VC) – the flow in a box heated on one side and cooled on the other side, is investigated using direct numerical simulations with Rayleigh numbers over the wide range of $10^7\le Ra\le 10^{14}$ and a fixed Prandtl number $Pr=10$ in a two-dimensional convection cell with unit aspect ratio. It is found that the dependence of the mean vertical centre temperature gradient $S$ on $Ra$ shows three different regimes: in regime I ($Ra \lesssim 5\times 10^{10}$), $S$ is almost independent of $Ra$; in the newly identified regime II ($5\times 10^{10} \lesssim Ra \lesssim 10^{13}$), $S$ first increases with increasing $Ra$ (regime $\textrm {{II}}_a$), reaches its maximum and then decreases again (regime $\textrm {{II}}_b$); and in regime III ($Ra\gtrsim 10^{13}$), $S$ again becomes only weakly dependent on $Ra$, being slightly smaller than in regime I. The transition from regime I to regime II is related to the onset of unsteady flows arising from the ejection of plumes from the sidewall boundary layers. The maximum of $S$ occurs when these plumes are ejected over approximately half of the area (downstream) of the sidewalls. The onset of regime III is characterized by the appearance of layered structures near the top and bottom horizontal walls. The flow in regime III is characterized by a well-mixed bulk region owing to continuous ejection of plumes over large fractions of the sidewalls, and, as a result of the efficient mixing, the mean temperature gradient in the centre $S$ is smaller than that of regime I. In the three different regimes, significantly different flow organizations are identified: in regime I and regime $\textrm {{II}}_a$, the location of the maximal horizontal velocity is close to the top and bottom walls; however, in regime $\textrm {{II}}_b$ and regime III, banded zonal flow structures develop and the maximal horizontal velocity now is in the bulk region. The different flow organizations in the three regimes are also reflected in the scaling exponents in the effective power law scalings $Nu\sim Ra^\beta$ and $Re\sim Ra^\gamma$. Here, $Nu$ is the Nusselt number and $Re$ is the Reynolds number based on maximal vertical velocity (averaged over vertical direction). In regime I, the fitted scaling exponents ($\beta \approx 0.26$ and $\gamma \approx 0.51$) are in excellent agreement with the theoretical predictions of $\beta =1/4$ and $\gamma =1/2$ for laminar VC (Shishkina, Phys. Rev. E., vol. 93, 2016, 051102). However, in regimes II and III, $\beta$ increases to a value close to 1/3 and $\gamma$ decreases to a value close to 4/9. The stronger $Ra$ dependence of $Nu$ is related to the ejection of plumes and the larger local heat flux at the walls. The mean kinetic dissipation rate also shows different scaling relations with $Ra$ in the different regimes.

We experimentally compared the drag-reduced turbulent channel flow of three different additives: a flexible polymer, a rigid polymer and a surfactant. A high drag reduction (HDR) of approximately 58 % was achieved using the flexible polymer, the rigid polymer and the surfactant. A maximum drag reduction (MDR) of approximately 70 % was also achieved using the flexible polymer and the surfactant. Solutions of flexible polymer and surfactant had a small shear viscosity, while the rigid polymer solution had a large shear viscosity with a considerable shear-thinning behaviour. The flexible polymer solution was the only fluid to exhibit a large extensional relaxation time. At HDR, the wall-normal distribution of mean velocity and the turbulent statistics of the drag-reduced flows were a function of the additive type and Reynolds number, Re. At MDR, the wall-normal distribution of mean velocity and turbulent statistics of the drag-reduced flows were similar, and not contingent on the additive type or Re. Due to its larger shear viscosity, the rigid polymer solution did not reach the MDR state in terms of drag reduction and mean velocity profile. However, the Reynolds stress profiles and turbulent length scale of the rigid polymer solution at HDR were similar to those of the flexible polymer and surfactant solutions at MDR. Our investigation demonstrated that different additives generate drag-reduced flows with similar turbulent statistics; however, no common rheological feature has been identified as of yet.

The unsteady flow separation and wake dynamics around finite wall-mounted circular cylinders fully immersed in a turbulent boundary layer (TBL) are investigated experimentally using a time-resolved particle image velocimetry (TR-PIV) system. The cylinder aspect ratios (h/d = 0.7–7.0, where h and d are the height and diameter of the cylinder, respectively) and the relative boundary layer thickness (δ/d = 8.7, where δ is the boundary layer thickness) were chosen to systematically investigate the effects of submergence ratio (δ/h = 1.2–12.4) using δ/h values much larger than that reported in the literature. With δ/h > 1.0, the cylinders encountered elevated turbulence levels (4%–10 %), reduced mean velocity and strong mean shear in the approach TBL which had profound effects on the attachment length and flapping motion of the reverse-flow region on the top surface of the cylinders. The time-averaged statistics including the mean velocities, Reynolds stresses and production terms were used to characterize the flow field and the large-scale anisotropy. The results showed that the wake structure of the submerged cylinders can be divided into dipoles and quadruples with a critical h/d = 3.5 and δ/h = 2.5. Both categories exhibited strong anisotropy, but the quadruples showed an interesting pattern where the streamwise Reynolds normal stress is less than the other components due to negative production in the wake region. Spectral analysis and joint-probability density functions are used to show that the reverse-flow region behind the cylinder is characterized by low-frequency flapping motions with a Strouhal number that decreases with increasing aspect ratio. The spatio-temporal evolution of the vortices also revealed the occurrence of cellular shedding behaviour where the vortices near the free end are shed discretely while those in the lower span are shed in the form of long streaky structures.

Estimation of the initial state of turbulent channel flow from limited data is investigated using an adjoint-variational approach. The data are generated from a reference direct numerical simulation that is subsampled at different spatiotemporal resolutions. When the velocity data are at 1/4096 the spatiotemporal resolution of the direct numerical simulation, the correlation coefficient between the true and adjoint-variational estimated state exceeds 99 %. The robustness of the algorithm to observation noise is demonstrated. In addition, the impact of the spatiotemporal density of the data on estimation quality is evaluated, and a resolution threshold is established for a successful reconstruction. The critical spanwise data resolution is proportional to the Taylor microscale, which characterizes the domain of dependence of an observation location. Owing to mean advection, either the streamwise or temporal data resolution must satisfy a criterion based on the streamwise Taylor microscale. A second configuration is considered where the subsampled data comprise velocities in the outer layer and wall shear stresses only. The near-wall flow statistics and coherent structures, although not sampled, are accurately reconstructed, which is possible because of the coupling between the outer flow and near-wall motions. Finally, the most challenging configuration is addressed where only the spatiotemporally resolved wall stresses are observed. The estimation remains accurate within the viscous sublayer and deteriorates significantly with distance from the wall. In wall units, this trend is nearly independent of the Reynolds number considered, and is indicative of the fundamental difficulty of reconstructing wall-detached motions from wall data.

The two-dimensional laminar flow of a viscous fluid induced by peristalsis due to a moving wall wave has been studied previously for a rectangular channel, a circular tube and a concentric circular annulus. Here, we study peristaltic flow in a non-axisymmetric annular tube: in this case, the flow is three-dimensional, with motions in the azimuthal direction. This type of geometry is motivated by experimental observations of the pulsatile flow of cerebrospinal fluid along perivascular spaces surrounding arteries in the brain, which is at least partially driven by peristaltic pumping due to pulsations of the artery. These annular perivascular spaces are often eccentric and the outer boundary is seldom circular: their cross-sections can be well matched by a simple, adjustable model consisting of an inner circle (the outer wall of the artery) and an outer ellipse (the outer edge of the perivascular space), not necessarily concentric. We use this geometric model as a basis for numerical simulations of peristaltic flow: the adjustability of the model makes it suitable for other applications. We concentrate on the general effects of the non-axisymmetric configuration on the flow and do not attempt to specifically model perivascular pumping. We use a finite-element scheme to compute the flow in the annulus driven by a propagating sinusoidal radial displacement of the inner wall. Unlike the peristaltic flow in a concentric circular annulus, the flow is fully three-dimensional: azimuthal pressure variations drive an oscillatory flow in and out of the narrower gaps, inducing an azimuthal wiggle in the streamlines. We examine the dependence of the flow on the elongation of the outer elliptical wall and the eccentricity of the configuration. We find that the time-averaged volumetric flow is always in the same direction as the peristaltic wave and decreases with increasing ellipticity or eccentricity. The additional shearing motion in the azimuthal direction will increase mixing and enhance Taylor dispersion in these flows, effects that might have practical applications.

We investigate vortex pair interactions at low Reynolds numbers. We base our analysis on the Q-criterion, where a vortex is defined as a region where the local rotation dominates the strain, and we make use of a topological approach to describe the qualitative changes of the vortex structure. In order to give a complete description of vortex pair interactions we further develop a general bifurcation theory for $Q$-vortices and prove that a threshold for vortex merging may occur when we allow two parameters to vary. To limit the number of free parameters, we study the interactions with two point vortices as the initial condition and show that the threshold is a codimension two bifurcation that appears as a cusp singularity on a bifurcation curve. We apply the general theory to the analytically tractable core growth model and conclude that a pair of co-rotating vortices merge only if their strength ratio, $\alpha =\varGamma _1/\varGamma _2$ is less than $4.58$. Below this threshold value, we observe two different regimes in which the merging processes can be described with different sequences of bifurcations. By comparison with Navier–Stokes simulations at different Reynolds numbers, we conclude that the merging threshold varies only slightly for Reynolds numbers up to $100$. Furthermore, we observe an excellent agreement between the core growth model and the numerical simulations for Reynolds numbers below 10. We therefore conclude that, instead of solving the Navier–Stokes equation numerically we can, for sufficiently small Reynolds numbers, apply the core growth model as a simple, analytically tractable model with a low dimension.

Interaction of polydisperse droplets in a turbulent air flow features prominently in a wide range of phenomena, such as warm rain initiation as an example. In the current study, we present an experimental investigation on the effects of initial droplet field characteristics on the maximum droplet size growth. By performing experiments in a vertically oriented air flow facility, the air flow turbulence was able to be controlled through the mean flow velocity and an active turbulence generator. The initial droplet field characteristics (droplet diameter range of 0–120 $\mathrm {\mu }$m) were varied using spray nozzles of different flow numbers. Based on quantitative measurements of the droplet size distribution at various spatial locations using phase Doppler interferometry (PDI), we estimated the droplet size growth rate $R$ as a function of turbulence intensity $I$, initial droplet number density $\rho _N$ and initial mean droplet size $\bar {D}$. For each ($\rho _N$, $\bar {D}$), we observed the occurrence of an optimum turbulence intensity $I^*$, with the corresponding maximum droplet size growth rate being $R^*$. Two different trends were observed. When $\rho _N$ and $\bar {D}$ were simultaneously increased and decreased, respectively, their competing influences resulted in small variations in $R^*$. In contrast, when $\bar {D}$ was held constant with a corresponding Stokes number $St$ smaller than unity, there existed a threshold $\rho _N$ above which $R^*$ increased rapidly with $\rho _N$. These trends were then understood through long-distance microscopy (LDM) measurements. Beyond the aforementioned threshold $\rho _N$, the fraction of uncorrelated small-sized $(St<1)$ droplet pairs was found to rapidly increase with $\rho _N$. Further detailed analysis of droplet tracking in the LDM images identified that the velocity fluctuations in the small-sized droplet pairs being induced by close encounters with inertial droplets was the underlying mechanism for the rapid increase of $R^*$ with $\rho _N$. This mechanism potentially explains how droplet collisions can be enhanced in small droplets if the droplet field is sufficiently polydisperse.

We investigate the flow of water in a two-dimensional laboratory aquifer recharged by artificial rainfall. As rainwater infiltrates, it forms a body of groundwater which can exit the aquifer only through one of its sides. The outlet, located high above the aquifer bottom, drives the flow upwards. Noting that the water table barely departs from the horizontal, we linearize the boundary condition at the free surface, and solve the flow equations in steady state. We find an approximate expression for the velocity potential, which accounts for the shape of the streamlines, and for the propagation of dye through the aquifer. Based on this theory, we calculate the travel time of water through the experiment, and find that its distribution decays exponentially, with a characteristic time that depends on the shape of the aquifer. We find that the hydrodynamic dispersion that occurs at the pore scale does not matter much for this distribution, which depends essentially on the geometry of the groundwater flow.

Dry lakes covered with a salt crust organised into beautifully patterned networks of narrow ridges are common in arid regions. Here, we consider the initial instability and the ultimate fate of buoyancy-driven convection that could lead to such patterns. Specifically, we look at convection in a deep porous medium with a constant throughflow boundary condition on a horizontal surface, which resembles the situation found below an evaporating salt lake. The system is scaled to have only one free parameter, the Rayleigh number, which characterises the relative driving force for convection. We then solve the resulting linear stability problem for the onset of convection. Further exploring the nonlinear regime of this model with pseudo-spectral numerical methods, we demonstrate how the growth of small downwelling plumes is itself unstable to coarsening, as the system develops into a dynamic steady state. In this mature state we show how the typical speeds and length scales of the convective plumes scale with forcing conditions, and the Rayleigh number. Interestingly, a robust length scale emerges for the pattern wavelength, which is largely independent of the driving parameters. Finally, we introduce a spatially inhomogeneous boundary condition – a modulated evaporation rate – to mimic any feedback between a growing salt crust and the evaporation over the dry salt lake. We show how this boundary condition can introduce phase locking of the downwelling plumes below sites of low evaporation, such as at the ridges of salt polygons.

We report experimental investigations on the mixing of a ferrofluid droplet with a non-magnetic miscible fluid in the presence of a time-dependent magnetic field on an open surface microfluidic platform. The bright-field visualization technique, in combination with micro-particle image velocimetry analysis, is carried out to explore the internal hydrodynamics of the ferrofluid droplet. Also, using the laser-induced fluorescence technique, we quantify the mass transfer occurring between the two droplets, which in effect, determines the underlying mixing performance under the modulation of the frequency of the applied magnetic field. We show that the magnetic nanoparticles exhibit complex spatio-temporal movements inside the ferrofluid droplet domain in a transient magnetic forcing environment, which, in turn, promotes the mixing efficiency in the convective mixing regime. Our analysis establishes that the movement of magnetic nanoparticles in the presence of the time-periodic field strengthens the flow instability, which initiates an augmented mixing in the present scenario. By performing numerical simulations, we also review the onset of instability phenomena, mainly stemming from the susceptibility mismatch between the magnetic and non-magnetic fluids. Inferences of the present analysis, which focuses on the simple, wireless, robust and low-cost open surface micromixing mechanism, will provide a potential solution for rapid droplet mixing without requiring a pH level or ion concentration dependency of the fluids.

The hydrodynamic response of an axisymmetric jet placed at various positions in a standing wave oriented normally to the jet is investigated. At the velocity and pressure nodes the axisymmetric ($m=0$) and first azimuthal ($m={\pm }1$) modes are excited, respectively, through manipulation of the jet exit boundary conditions. At positions between the nodes, both the $m=0$ and $m={\pm }1$ modes are simultaneously excited resulting in asymmetric forcing due to the phase difference between the transverse and longitudinal acoustic fluctuations. This leads to the asymmetric formation of vortices in the near field and bifurcation into two or more momentum streams further downstream. The dominant momentum stream is deflected in the direction of the velocity node. It is shown that the asymmetric response can be well approximated by a superposition of the boundary conditions at the pressure and velocity nodes where the contributions from each mode are proportional to the acoustic pressure and velocity. A method is proposed to characterize the bifurcation behaviour statistically via moments of the probability density functions constructed from profiles of streamwise momentum. The jet symmetry and momentum spreading are shown to be proportional to the magnitude of the transverse acoustic velocity. Finally, the streamwise velocity is reconstructed as a superposition of Gaussian profiles providing a robust method to characterize the number of individual momentum streams which also shows that each of the streams behave self-similarly.

The aeroacoustic source mechanism of a deep rectangular cavity, which has an aspect ratio of $D/L=2.632$ and is subjected to a turbulent boundary layer of $\theta /L=0.0345$ at a Mach number of 0.2, is investigated by using a high-order accurate large-eddy simulation. The primary aim of this study is to provide an improved understanding of the fluid–acoustic coupling mechanism that triggers a self-sustained acoustic resonance in a deep cavity. Various analysis methods, which include Doak's momentum potential theory that allows for the separation of hydrodynamic and acoustic components, are used to provide highly detailed investigations and findings. The vortex dynamics near the cavity opening region is investigated as the potential primary source of noise generation. In addition, the noise generation mechanism is quantitatively explained by the onset of the separation region near the downstream corner that ensues from the synchronised shear layer–wall interaction. The current work extensively focuses on the fluid–acoustic coupling mechanism, and it is found that the acoustic resonance favourably modulates the hydrodynamic fluctuation near the upstream corner of the cavity. Furthermore, the current study also suggests that nonlinear interactions between fundamental acoustic resonance and higher harmonics are plausible. Based on the discussions provided in this paper, a semi-empirical model to predict the critical free stream velocity at which a strong fluid–acoustic coupling occurs as a function of cavity geometry and inflow boundary-layer property is proposed.

Koopman decomposition is a nonlinear generalization of eigen-decomposition, and is being increasingly utilized in the analysis of spatio-temporal dynamics. Well-known techniques such as the dynamic mode decomposition (DMD) and its linear variants provide approximations to the Koopman operator, and have been applied extensively in many fluid dynamic problems. Despite being endowed with a richer dictionary of nonlinear observables, nonlinear variants of the DMD, such as extended/kernel dynamic mode decomposition (EDMD/KDMD) are seldom applied to large-scale problems primarily due to the difficulty of discerning the Koopman-invariant subspace from thousands of resulting Koopman eigenmodes. To address this issue, we propose a framework based on a multi-task feature learning to extract the most informative Koopman-invariant subspace by removing redundant and spurious Koopman triplets. In particular, we develop a pruning procedure that penalizes departure from linear evolution. These algorithms can be viewed as sparsity-promoting extensions of EDMD/KDMD. Furthermore, we extend KDMD to a continuous-time setting and show a relationship between the present algorithm, sparsity-promoting DMD and an empirical criterion from the viewpoint of non-convex optimization. The effectiveness of our algorithm is demonstrated on examples ranging from simple dynamical systems to two-dimensional cylinder wake flows at different Reynolds numbers and a three-dimensional turbulent ship-airwake flow. The latter two problems are designed such that very strong nonlinear transients are present, thus requiring an accurate approximation of the Koopman operator. Underlying physical mechanisms are analysed, with an emphasis on characterizing transient dynamics. The results are compared with existing theoretical expositions and numerical approximations.

Wave radiation and diffraction problems of a body floating in an open water channel confined by two semi-infinite ice sheets are considered. The linearized velocity potential theory is used for fluid flow and a thin elastic plate model is adopted for the ice sheet. The Green function, which satisfies all the boundary conditions apart from that on the body surface, is first derived. This is obtained through applying Fourier transform in the longitudinal direction of the channel, and matched eigenfunction expansions in the transverse plane. With the help of the derived Green function, the boundary integral equation of the potential is derived and it is shown that the integrations over all other boundaries, including the bottom of the fluid, free surface, ice sheet, ice edge as well as far field will be zero, and only the body surface has to be retained. This allows the problem to be solved through discretization of the body surface only. Detailed results for hydrodynamic forces are provided, which are generally highly oscillatory owing to complex wave–body–channel interaction and body–body interaction. In depth investigations are made for the waves confined in a channel, which does not decay at infinity. Through this, a detailed analysis is presented on how the wave generated by a body will affect the other bodies even when they are far apart.